Mixed perverse sheaves for schemes over number fields

This paper generalizes the definition of mixed perverse sheavesto schemes of finite type over a number field.Their basic properties, e.g., characterization of simple objects, are shown.     

On the Schwartz space of the basic affine space

Let G be the group of points of a split reductive algebraic group G over a local field k and let X = G / U where U is the group of k-points of a maximal unipotent subgroup of G. In this paper we construct a certain canonical G-invariant space (called the Schwartz space of X) of functions on X, which is an extension of the space of smooth compactly supported functions on X. We show that the space of all elements of , which are invariant under the Iwahori subgroup I of G, coincides with the space generated by the elements of the so called periodic Lusztig basis, introduced recently by G. Lusztig (cf. [10] and [11]). We also give an interpretation of this space in terms of a certain equivariant K-group (this was also done by G. Lusztig — cf. [12]). Finally we present a global analogue of , which allows us to give a somewhat non-traditional treatment of the theory of the principal Eisenstein series.     

On the structure of rigid semistable sheaves on algebraic surfaces

LetS be a smooth projective surface, letK be the canonical class ofS and letH be an ample divisor such thatH • K < 0. We prove that for any rigid sheafF (Ext¹ (F, F) = 0) that is Mumford-Takemoto semistable with respect toH there exists an exceptional set (E ₁ ,..., E _n ) of sheaves onS such thatF can be constructed from {E _i } by means of a finite sequence of extensions. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 692–700, November, 1998. The author wishes to express his gratitude to S. A. Kuleshov for useful discussions and to A. N. Rudakov and A. L. Gorodentsev for their attention to the present work. This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-01323 and by the INTAS Foundation.     

On the reducibility of characteristic varieties

We show that some monodromies in the Morse local systems of a conically stratified perverse sheaf imply that other Morse local systems for smaller strata do not vanish. This result is then used to explain the examples of reducible characteristic varieties of Schubert varieties given by Kashiwara and Saito in type and by Boe and Fu for the Lagrangian Grassmannian.

     

Euler number of the moduli space of sheaves on a rational nodal curve

In this paper, we use finite group actions to compute the Euler number of the moduli space of rank 2 stable sheaves on a rational nodal curve.

     

On the ‐action on G‐semistable locally free sheaves on

We show that the divisor of jumping lines of any , the moduli space of Gieseker‐semistable locally free sheaves of rank 2 on with , is reduced for . By a lemma of Artamkin this implies, that there are exactly ‐orbits in , the subset of those , which are trivial at a certain line .     

Positive Harris recurrence and exponential ergodicity of the basic affine jump-diffusion

In this article, we find the transition densities of the basic affine jump-diffusion (BAJD), which has been introduced by Duffie and Gârleanu as an extension of the CIR model with jumps. We prove the positive Harris recurrence and exponential ergodicity of the BAJD. Furthermore, we prove that the unique invariant probability measure π of the BAJD is absolutely continuous with respect to the Lebesgue measure and we also derive a closed-form formula for the density function of π.     

Gluing Affine 2-Manifolds with Polygons

Affine structures on surfaces are constructed by gluing polygons. The geometry of the affine surface depends on the shape of the polygon(s) and the particular gluing transformations used. The affine version of the Poincaré fundamental polygon theorem expresses the fundamental group and holonomy of the surface in terms of the gluing data. The theorem may be used to construct all complete affine structures on the 2-torus. The space of inequivalent holonomy representations of such structures is homeomorphic to R2.     

Character sheaves on the semi-stable locus of a group compactification

We study the intermediate extension of the character sheaves on an adjoint group to the semi-stable locus of its wonderful compactification. We show that the intermediate extension can be described by a direct image construction. As a consequence, we show that the “ordinary” restriction of a character sheaf on the compactification to a “semi-stable stratum” is a shift of semisimple perverse sheaf and is closely related to Lusztig's restriction functor (from a character sheaf on a reductive group to a direct sum of character sheaves on a Levi subgroup). We also provide a (conjectural) formula for the boundary values inside the semi-stable locus of an irreducible character of a finite group of Lie type, which gives a partial answer to a question of Springer (2006) [21]. This formula holds for Steinberg character and characters coming from generic character sheaves. In the end, we verify Lusztig's conjecture Lusztig (2004) [16, 12.6] inside the semi-stable locus of the wonderful compactification.     

Bounds on affine caps

We present a short and elementary proof for an upper bound on caps in affine spaces. The bound generalizes and strengthens a result by Meshulam. The proof is inspired by Meshulam's Fourier transform method without making explicit use of the Fourier transform. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 111–115, 2002; DOI 10.1002/jcd.10000     

利用有限仿射几何构作Cartesian认证码

设 0 相似文献   

Picard groups of the moduli spaces of semistable sheaves I

We compute the Picard group of the moduli spaceU′ of semistable vector bundles of rankn and degreed on an irreducible nodal curveY and show thatU′ is locally factorial. We determine the canonical line bundles ofU′ andU _L ′, the subvariety consisting of vector bundles with a fixed determinant. For rank 2, we compute the Picard group of other strata in the compactification ofU′.     

Moduli of sheaves and the Chow group of K3 surfaces

Let X be a projective complex K  3 surface. Beauville and Voisin singled out a 0-cycle _cX$c_X$ on X of degree 1 and Huybrechts proved that the second Chern class of a rigid simple vector-bundle on X   is a multiple of _cX$c_X$ if certain hypotheses hold. We believe that the following generalization of Huybrechts? result holds. Let M be a moduli space of stable pure sheaves on X with fixed cohomological Chern character: the set whose elements are second Chern classes of sheaves parametrized by the closure of M (in the corresponding moduli spaces of semistable sheaves) depends only on the dimension of M. We will prove that the above statement holds under some additional assumptions on the Chern character.     

Diagrammatic description for the categories of perverse sheaves on isotropic Grassmannians

For each integer \(k\ge 4\), we describe diagrammatically a positively graded Koszul algebra \(\mathbb {D}_k\) such that the category of finite dimensional \(\mathbb {D}_k\)-modules is equivalent to the category of perverse sheaves on the isotropic Grassmannian of type \(\mathrm{D}_k\) or \(\mathrm{B}_{k-1}\), constructible with respect to the Schubert stratification. The algebra is obtained by a (non-trivial) “folding” procedure from a generalized Khovanov arc algebra. Properties such as graded cellularity and explicit closed formulas for graded decomposition numbers are established by elementary tools.     

Some remarks on tautological sheaves on Hilbert schemes of points on a surface

We study tautological sheaves on the Hilbert scheme of points on a smooth quasi-projective algebraic surface by means of the Bridgeland–King–Reid transform. We obtain Brion–Danila’s Formulas for the derived direct image of tautological sheaves or their double tensor product for the Hilbert–Chow morphism; as an application we compute the cohomology of the Hilbert scheme with values in tautological sheaves or in their double tensor product, thus generalizing results previously obtained for tautological bundles.      

Nonlinearizable actions of dihedral groups on affine space

Let be a reductive, non-abelian, algebraic group defined over . We investigate algebraic -actions on the total spaces of non-trivial algebraic -vector bundles over -modules with great interest in the case that is a dihedral group. We construct a map classifying such actions of a dihedral group in some cases and describe the spaces of those non-linearizable actions in some examples.